**NOTE: this assignment will be turned in on APRIL 10 (before the WPR), and count as part of the WPR grade.**

*In this assignment, you'll be investigating the definition of distance between compact sets. There are a few equivalent ways to define this distance (see Problem 14 in Chapter 2) for another example. This homework assignment requires a review of the definition of continuity, and provides an example of using the Extreme Value Theorem.*

# Distance Between a Point and a Set

**1.** Let *x* be a point in a metric space *M*, and let $S\subset M$. Prove that the function $f:S\to\mathbb{R}$ defined by $f(y)=d(x,y)$ is continuous. You should use the epsilon-delta definition of continuity. *(Hint: You'll need the triangle inequality.)*

**2.** Now suppose that $S\subset M$ is compact and *x* is still a point in the metric space. Prove that some point $s\in S$ is "closest" to *x*. In other words, show that there exists some $s\in S$ such that $d(s,x)\leq d(y,x)$ for all $y\in S$. *(Hint: Use one of the "big theorems" that we've been discussing in class.)*

**3.** By the previous problem, we can define the distance $d(x,S)$ between a point and a set to be the distance between *x* and the point in *S* which is *closest to* *x*. When is $d(x,S)=0$?

**4.** Suppose the metric space is chosen to be $\mathbb{R}$. show that if *S* is not closed, there may be no minimum distance between a point *x* and *S*. What if *S* is closed but not bounded?

# Distance Between Sets

**5.** Given that the function $f(y)=d(x,y)$ is continuous, as proven in Problem 1, show that the function $g(x)=d(x,S)$ is continuous.

**6.** Suppose that *S* and *T* are compact subsets of *M*. One can define $g:T\to\mathbb{R}$ by $g(x)=d(x,S)$. Prove that *g* obtains a minimum value. What are the implications for the sets *S* and *T*?

**7.** Hence, we can define $d(S,T)$. Show that $diam(S\cup T)\leq diam(S)+diam(T)+d(S,T)$. Provide an example (easiest in $\mathbb{R}$) showing that equality is possible.